 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-b/35/a/solution

The operator

\hat{J}_{z}

describes the

z

component of angular momentum, whereas

\hat{J}^{2}

describes its total squared magnitude. For a spherical harmonic

Y_{J M}

the corresponding eigenvalues are

\hat{J}_{z} Y_{J M} = M ℏ Y_{J M}, \hat{J}^{2} Y_{J M} = J (J + 1) ℏ^{2} Y_{J M} .

Thus the magnitude is

ℏ J (J + 1)

, not

J ℏ

. For ordinary orbital rotation

J = 0, 1, 2, \dots

and

M = - J, - J + 1, \dots, J

. The simultaneous values are possible because these two operators commute.

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